Problem: Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{y^2 - 6y}{y^2 - 12y + 36}$
First factor the expressions in the numerator and denominator. $ \dfrac{y^2 - 6y}{y^2 - 12y + 36} = \dfrac{(y)(y - 6)}{(y - 6)(y - 6)} $ Notice that the term $(y - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y - 6)$ gives: $a = \dfrac{y}{y - 6}$ Since we divided by $(y - 6)$, $y \neq 6$. $a = \dfrac{y}{y - 6}; \space y \neq 6$